9 MEASURABLE FUNCTIONS 145 (b) A sequence (wn) of finite partitions of X is said to be fundamental if wn+1 is finer than wn for ail n and if the maximum of the diameters of the sets of rnn tends to 0 as n tends to + oo. Show that if/is a bounded integrable function on X, then there exists at least one fundamental sequence (mn) of finite partitions of X, consisting of integrable sets, and such that the sequences (swn(f)) and (SWn(/)) both tend to j fd(JL. (c) Let/be bounded and continuous almost everywhere on X (Problem 6). Show that, for all fundamental sequences (wn) of finite partitions of X consisting of integrable sets, the sequences ($*,„(/)) and (SWn(/)) both tend to fdyu. (Observe that if An is the closed set of points x e X at which the oscillation of/(Section 3.14) is ^l//i, then ^(An) tends to 0 as n tends to + oo; each An has an open neighborhood Vn such that the points of Vn have a distance <l/n from An and such that ju-(Vn) tends to 0. For a partition wk whose sets all have diameter <l/n, consider separately the sets of the partition which meet Vn and those which do not.) If/is bounded and lower semi- continuous on X, show that smn(f) tends to /dp, for every fundamental sequence (run) of finite partitions of X consisting of integrable sets. (d) A subset A of X is said to be quadrable (with respect to /x) if its characteristic function <pA is continuous almost everywhere, or equivalently if the frontier of A is jLc-negligible. Show that every point jc0 in X has a fundamental system of quad- rable open neighborhoods. (For each neighborhood V of x0, let /: X-> [0,1] be a continuous mapping such that f(x0) = 1 and f(x) — 0 for all x e X — V. For each a e ]0,1 [, consider the set of points x e X such that f(x) > a.) Deduce that there exists a fundamental sequence of finite partitions of X consisting of sets which are either open or negligible. Give an example of a nonquadrable closed set (Section 13.8, Problem 4(a); cf. Section 13.21, Problem 2). (e) Let (wn) be a fundamental sequence of finite partitions of X consisting of sets which are either open or negligible. If/is a bounded function on X, let g be the largest lower semicontinuous function ^/; show that smn(f) = Svn(g). Deduce that the sequences (smn(f)) and (SWn(/)) tend to the same limit if and only if/is continuous almost everywhere. (f) Use (e) to give an example of a negligible function/and a fundamental sequence (utn) of finite partitions of X consisting of integrable sets, such that the sequences (•?«*„(/)) and (SOTn(/)) do not tend to the same limit. 8. Let/be a measurable mapping of X into a complete metric space E, and let K be a compact subset of X. Show that f\ K can be approximated uniformly by measurable step-functions if and only if/(K) is relatively compact in E. 9. Suppose that X is compact. Let (/„) be a sequence of ju-measurable finite real-valued functions on X. Show that the following properties are equivalent: (1) There exists a subsequence of (/,) which tends to 0 almost everywhere in X. (2) There exists a sequence (tn) of finite real numbers, such that lim sup |/n| > 0 n-foo and such that the series with general term tnfn(x) converges almost everywhere in X. (3) There exists a sequence (tn) of finite real numbers such that £ |fB| = -|- oo and such that the series with general term tnfn(x) converges absolutely almost every- where in X." MarkofT-Kakutani theorem")-